Quantization effects for a fourth-order equation of exponential growth in dimension $4$

Abstract

We investigate the asymptotic behaviour as $k\to+\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\varOmega)$ of solutions of the equations $\Delta^2u_k=V_k\mathrm{e}^{4u_k}$ on $\varOmega$, where $\varOmega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to+\infty}V_k=1$ in $C^0_{\mathrm{loc}}(\varOmega)$. The corresponding two-dimensional problem was studied by Brézis and Merle and Li and Shafrir, who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Robert and Struwe in 2006, such a quantization does not hold in dimension $4$ for the problem in its full generality. We prove here that, under a natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension $2$.